chapter 12: compressible flow -...

Chapter 12: Compressible Flow

Eric G. PatersonDepartment of Mechanical and Nuclear Engineering

The Pennsylvania State University

Spring 2005

Chapter 12: Compressible FlowME33 : Fluid Flow 2

Note to Instructors

These slides were developed1, during the spring semester 2005, as a teaching aid

for the undergraduate Fluid Mechanics course (ME33: Fluid Flow) in the Department of

Mechanical and Nuclear Engineering at Penn State University. This course had two

sections, one taught by myself and one taught by Prof. John Cimbala. While we gave

common homework and exams, we independently developed lecture notes. This was

also the first semester that Fluid Mechanics: Fundamentals and Applications was

used at PSU. My section had 93 students and was held in a classroom with a computer,

projector, and blackboard. While slides have been developed for each chapter of Fluid

Mechanics: Fundamentals and Applications, I used a combination of blackboard and

electronic presentation. In the student evaluations of my course, there were both positive

and negative comments on the use of electronic presentation. Therefore, these slides

should only be integrated into your lectures with careful consideration of your teaching

style and course objectives.

Eric Paterson

Penn State, University Park

August 2005

1 This Chapter was not covered in our class. These slides have been developed at the request of McGraw-Hill


Objectives

Appreciate the consequences of compressibility in gas flows

Understand why a nozzle must have a diverging section to accelerate a gas to supersonic speeds

Predict the occurrence of shocks and calculate property changes across a shock wave

Understand the effects of friction and heat transfer on compressible flows


Stagnation Properties

Recall definition of enthalpy

which is the sum of internal

energy u and flow energy P/ρ

For high-speed flows,

enthalpy and kinetic energy

are combined into

stagnation enthalpy h0



Steady adiabatic flow through duct with no shaft/electrical work and no change in elevation and potential energy

Therefore, stagnation enthalpy remains constant during steady-flow process



If a fluid were brought to a complete stop (V2 = 0)

Therefore, h0 represents the enthalpy of a fluid when it is brought to rest adiabatically.

During a stagnation process, kinetic energy is

converted to enthalpy.

Properties at this point are called stagnation

properties (which are identified by subscript 0)



If the process is also reversible, the stagnation state is called the isentropic stagnation state.

Stagnation enthalpy is the same for isentropic and actual stagnation states

Actual stagnation pressure P0,act is lower than P0 due to increase in entropy s as a result of fluid friction.

Nonetheless, stagnation processes are often approximated to be isentropic, and isentropic properties are referred to as stagnation properties



For an ideal gas, h = CpT, which allows the h0 to be

rewritten

T0 is the stagnation temperature. It represents the

temperature an ideal gas attains when it is brought to rest

adiabatically.

V2/2Cp corresponds to the temperature rise, and is called the

dynamic temperature

For ideal gas with constant specific heats, stagnation

pressure and density can be expressed as



When using stagnation enthalpies, there is no

need to explicitly use kinetic energy in the

energy balance.

Where h01 and h02 are stagnation enthalpies at

states 1 and 2.

If the fluid is an ideal gas with constant specific

heats


Speed of Sound and Mach Number

Important parameter in compressible flow is the speed of sound.

Speed at which infinitesimally small pressure wave travels

Consider a duct with a moving piston

Creates a sonic wave moving to the right

Fluid to left of wave front experiences incremental change in properties

Fluid to right of wave front maintains original properties



Construct CV that encloses

wave front and moves with it

Mass balance

cancel Neglect H.O.T.



Energy balance ein = eout

cancel cancel Neglect H.O.T.



Using the thermodynamic relation

Combing this with mass and energy

conservation gives

For an ideal gas



Since

R is constant

k is only a function of T

Speed of sound is only

a function of temperature



Second important

parameter is the

Mach number Ma

Ratio of fluid velocity

to the speed of sound

Flow regimes

classified in terms of

Ma

Ma < 1 : Subsonic

Ma = 1 : Sonic

Ma > 1 : Supersonic

Ma >> 1 : Hypersonic

Ma ≈ 1 : Transonic


One-Dimensional Isentropic Flow

For flow through

nozzles, diffusers, and

turbine blade passages,

flow quantities vary

primarily in the flow

direction

Can be approximated as

1D isentropic flow

Consider example of

Converging-Diverging

Duct


One-Dimensional Isentropic Flow

Example 12-3 illustrates

Ma = 1 at the location of the

smallest flow area, called the

throat

Velocity continues to increase

past the throat, and is due to

decrease in density

Area decreases, and then

increases. Known as a

converging - diverging

nozzle. Used to accelerate

gases to supersonic speeds.


One-Dimensional Isentropic Flow Variation of Fluid Velocity with Flow Area

Relationship between V, ρ, and A are complex

Derive relationship using continuity, energy,

speed of sound equations

Continuity

Differentiate and divide by mass flow rate (ρAV)



Derived relation (on

image at left) is the

differential form of

Bernoulli’s equation.

Combining this with result

from continuity gives

Using thermodynamic

relations and rearranging



This is an important relationship

For Ma < 1, (1 - Ma2) is positive ⇒ dA and dP have

the same sign.

Pressure of fluid must increase as the flow area of the duct

increases, and must decrease as the flow area decreases

For Ma > 1, (1 - Ma2) is negative ⇒ dA and dP have

opposite signs.

Pressure must increase as the flow area decreases, and

must decrease as the area increases



A relationship between dA and dV can be

derived by substituting ρV = -dP/dV (from the

differential Bernoulli equation)

Since A and V are positive

For subsonic flow (Ma < 1) dA/dV < 0

For supersonic flow (Ma > 1) dA/dV > 0

For sonic flow (Ma = 1) dA/dV = 0



Comparison of flow properties in subsonic and supersonic nozzles and diffusers


One-Dimensional Isentropic Flow Property Relations for Isentropic Flow of Ideal Gases

Relations between static properties and stagnation properties in

terms of Ma are useful.

Earlier, it was shown that stagnation temperature for an ideal gas

was

Using definitions, the dynamic temperature term can be expressed

in terms of Ma



Substituting T0/T ratio into P0/P and ρ0/ρrelations (slide 8)

Numerical values of T0/T, P0/P and ρ0/ρcompiled in Table A-13 for k=1.4

For Ma = 1, these ratios are called critical ratios


Isentropic Flow Through Nozzles

Converging or converging-diverging nozzles are

found in many engineering applications

Steam and gas turbines, aircraft and spacecraft

propulsion, industrial blast nozzles, torch nozzles

Here, we will study the effects of back pressure

(pressure at discharge) on the exit velocity,

mass flow rate, and pressure distribution along

the nozzle


Isentropic Flow Through NozzlesConverging Nozzles

State 1: Pb = P0, there is no

flow, and pressure is constant.

State 2: Pb < P0, pressure along

nozzle decreases.

State 3: Pb =P* , flow at exit is

sonic, creating maximum flow

rate called choked flow.

State 4: Pb < Pb, there is no

change in flow or pressure

distribution in comparison to

state 3

State 5: Pb =0, same as state 4.



Under steady flow conditions, mass flow rate is constant

Substituting T and P from the expressions on slides 23 and 24 gives

Mass flow rate is a function of stagnation properties, flow area, and Ma



The maximum mass flow rate through a nozzle with a given throat area A* is fixed by the P0 and T0 and occurs at Ma = 1

This principal is important for chemical processes, medical devices, flow meters, and anywhere the mass flux of a gas must be known and controlled.


Isentropic Flow Through NozzlesConverging-Diverging Nozzles

The highest velocity in a converging nozzle

is limited to the sonic velocity (Ma = 1),

which occurs at the exit plane (throat) of the

nozzle

Accelerating a fluid to supersonic velocities

(Ma > 1) requires a diverging flow section

Converging-diverging (C-D) nozzle

Standard equipment in supersonic aircraft and

rocket propulsion

Forcing fluid through a C-D nozzle does not

guarantee supersonic velocity

Requires proper back pressure Pb



1. P0 > Pb > Pc

Flow remains subsonic, and

mass flow is less than for choked flow. Diverging section

acts as diffuser

2. Pb = PC

Sonic flow achieved at throat.

Diverging section acts as

diffuser. Subsonic flow at exit. Further decrease in Pb has no

effect on flow in converging

portion of nozzle



3. PC > Pb > PE

Fluid is accelerated to supersonic velocities in the diverging section as the pressure decreases. However, acceleration stops at location of normal shock. Fluid decelerates and is subsonic at outlet. As Pb is decreased, shock approaches nozzle exit.

4. PE > Pb > 0Flow in diverging section is supersonic with no shock forming in the nozzle. Without shock, flow in nozzle can be treated as isentropic.


Shock Waves and Expansion Waves

Review

Sound waves are created by small pressure disturbances and travel at the speed of sound

For some back pressures, abrupt changes in fluid properties occur in C-D nozzles, creating a shock wave

Here, we will study the conditions under which shock waves develop and how they affect the flow.


Shock Waves and Expansion WavesNormal Shocks

Shocks which occur in a plane normal to the direction of flow are called normal shock waves

Flow process through the shock wave is highly irreversible and cannot be approximated as being isentropic

Develop relationships for flow properties before and after the shock using conservation of mass, momentum, and energy



Conservation of mass

Conservation of energy

Conservation of momentum

Increase in entropy



Combine conservation of mass and energy into a single equation and plot on h-sdiagram

Fanno Line : locus of states that have the same value of h0 and mass flux

Combine conservation of mass and momentum into a single equation and plot on h-sdiagram

Rayleigh line

Points of maximum entropy correspond to Ma = 1.

Above / below this point is subsonic / supersonic



There are 2 points where the Fanno and Rayleigh lines intersect : points where all 3 conservation equations are satisfied

Point 1: before the shock (supersonic)

Point 2: after the shock (subsonic)

The larger Ma is before the shock, the stronger the shock will be.

Entropy increases from point 1 to point 2 : expected since flow through the shock is adiabatic but irreversible



Equation for the Fanno line for an ideal gas with constant specific heats can be derived

Similar relation for Rayleigh line is

Combining this gives the intersection points


Shock Waves and Expansion WavesOblique Shocks

Not all shocks are normal to flow direction.

Some are inclined to the flow direction, and are called oblique shocks



At leading edge, flow is deflected through an angle θ called the turning angle

Result is a straight oblique shock wave aligned at shock angle βrelative to the flow direction

Due to the displacement thickness, θ is slightly greater than the wedge half-angle δ.



Like normal shocks, Ma decreases across the oblique shock, and are only possible if upstream flow is supersonic

However, unlike normal shocks in which the downstream Ma is always subsonic, Ma2 of an oblique shock can be subsonic, sonic, or supersonic depending upon Ma1 and θ.



All equations and

shock tables for

normal shocks apply

to oblique shocks as

well, provided that we

use only the normal

components of the

Mach number

Ma1,n = V1,n/c1

Ma2,n = V2,n/c2

θ−β-Ma relationship



If wedge half angle θ

> θmax, a detached

oblique shock or bow

wave is formed

Much more

complicated that

straight oblique

shocks.

Requires CFD for

analysis.



Similar shock waves see for axisymmetric

bodies, however, θ−β-Ma relationship and

resulting diagram is different than for 2D bodies



For blunt bodies,

without a sharply

pointed nose, δ = 90°,

and an attached

oblique shock cannot

exist regardless of

Ma.


Shock Waves and Expansion WavesPrandtl-Meyer Expansion Waves

In some cases, flow is turned in the opposite direction across the shock

Example : wedge at angle of attack θgreater than wedge half angle δ

This type of flow is called an expanding flow, in contrast to the oblique shock which creates a compressing flow.

Instead of a shock, a expansion fanappears, which is comprised of infinite number of Mach waves called Prandtl-Meyer expansion waves

Each individual expansion wave is isentropic : flow across entire expansion fan is isentropic

Ma2 > Ma1P, ρ, T decrease across the fan

Flow turns gradually as each successful Mach wave turns

the flow ay an infinitesimal amount



Prandtl-Meyer expansion fans also occur in axisymmetric flows, as in the corners and trailing edges of the cone cylinder.



Interaction between shock waves and expansions waves in “over expanded” supersonic jet


Duct Flow with Heat Transfer

and Negligible Friction

Many compressible flow problems encountered in practice involve chemical reactions such as combustion, nuclear reactions, evaporation, and condensation as well as heat gain or heat loss through the duct wall

Such problems are difficult to analyze

Essential features of such complex flows can be captured by a simple analysis method where generation/absorption is modeled as heat transfer through the wall at the same rate

Still too complicated for introductory treatment since flow may involve friction, geometry changes, 3D effects

We will focus on 1D flow in a duct of constant cross-sectional area with negligible frictional effects




Consider 1D flow of an ideal gas with constant cp through a duct with constant A with heat transfer but negligible friction (known as Rayleigh flow)

Continuity equation

X-Momentum equation




Energy equationCV involves no shear, shaft, or other forms of work, and potential energy change is negligible.

For and ideal gas with constant cp, ∆h = cp∆T

Entropy changeIn absence of irreversibilities such as friction, entropy changes by heat transfer only




Infinite number of downstream states 2 for a given upstream state 1

Practical approach is to assume various values for T2, and calculate all other properties as well as q.

Plot results on T-s diagram

Called a Rayleigh line

This line is the locus of all physically attainable downstream states

S increases with heat gain to point a which is the point of maximum entropy (Ma =1)


Adiabatic Duct Flow with Friction

Friction must be included for flow through

long ducts, especially if the cross-sectional

area is small.

Here, we study compressible flow with

significant wall friction, but negligible heat

transfer in ducts of constant cross section.



Consider 1D adiabatic flow of an ideal gas with constant cp through a duct with constant A with significant frictional effects (known as Fanno flow)

Continuity equation

X-Momentum equation




Energy equationCV involves no heat or work, and potential energy change is negligible.

For and ideal gas with constant cp, ∆h = cp∆T

Entropy changeIn absence of irreversibilities such as friction, entropy changes by heat transfer only




Infinite number of downstream states 2 for a given upstream state 1

Practical approach is to assume various values for T2, and calculate all other properties as well as friction force.

Plot results on T-s diagram

Called a Fanno line

This line is the locus of all physically attainable downstream states

s increases with friction to point of maximum entropy (Ma =1).

Two branches, one for Ma < 1, one for Ma >1

chapter 12: compressible flow -...

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